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One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Turbomachinery & Turbulence.Lecture 4: Design and analysis of an axial-flow compression
stage.
F. Ravelet
Laboratoire DynFluid, Arts et Metiers-ParisTech
February 7, 2016
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Fundamental law of turbomachines
Moment of momentum
Fluid enters at a flow rate m at r1 withtangential velocity Cθ1.
It leaves the control volume at r2 withtangential velocity Cθ2.
Moment of momentum, steady version,along a streamline:
τa = m (r2Cθ2 − r1Cθ1)
Power:
τaω = m (U2Cθ2 − U1Cθ1)
Link to energy exchange (steady process,adiabatic):
∆h0 = ∆ (UCθ)
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Fundamental law of turbomachines
Rothalpy
Along a streamline, the quantity calledrothalpy is constant:
I = h0 − UCθ = cte
Using the velocity triangle:
I = h +W 2
2−
U2
2= cte
Different contributions:
∆h0 = ∆ (UWθ) + ∆(U2
)Aerodynamic forces work + Coriolis forces(2~ω × ~W ) work.
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Axial-flow stage
Axial-flow compression stage
Axial-flow stage: U = cte along a streamline. If one assumes Cm = cte:
∆h0 = U (Wθ2 −Wθ1)
= UCm (tanβ2 − tanβ1)
Watch out: β < 0. For a compression, |Wθ2| < |Wθ1| ⇒ h2 > h1.
Stator: h0 = cte but |C3| < |C2| ⇒ h3 > h2.
Conversion of kinetic energy to pressure (and degradation to internal energy).
The relative flow is decelerated in the rotor. The absolute flow is decelerated inthe stator. Diffusion (adverse pressure gradient) limits deflection to 40o .
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Axial-flow stage
Mechanical energy loss
Mollier diagram for a steady blade cascade h02 = h01.
Losses are related to ∆p0:
ˆT0ds0 = −
ˆdp0
ρ0
∆f 'p02 − p01
ρ01
Losses with respect to isentropic arerelated to kinetic energy:
Is. Loss =1
2
(C2
2s − C22
)
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
2D Blade cascade
Blade cascade
∆h0 = UCm (tanβ2 − tanβ1)
Work depends on flow deflection ∆β.
Two-dimensional profile:
l : chord lengtht: thickness of the profileθ: camber angle.
α′1,2: blade inlet (outlet) angle
Two-dimensional cascade:
ξ: stagger angleσ = l/s: solidityα1,2: inlet (outlet) flow angle
i = α1 − α′1: incidence angle
δ = α2 − α′2: deviation
ε = α2 − α1: deflectionaoa = α1 − ξ: angle of attack
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
2D Blade cascade
Cascade characteristics
Blade profiles: a certain thickness distribution(NACA65, British C series,...)
Cascade characteristics: for given profiles,stagger angle and solidity,
as a function of α1, M1, Re1:
Exit flow angle α2
stagnation pressure loss coefficient YP
The results are also presented as ε as a function of
aoa, as lift and drag coefficient or as energy loss
coefficient ζ.
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
2D Blade cascade
Losses
Stagnation pressure loss coefficient(compression):
Yp =p01 − p02
p01 − p1
Energy loss coefficient:
ζ =
(C 2
2s − C 22
)C 2
1
Yp = ζ for M → 0.
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
2D Blade cascade
Cascade parameters measurements
Measurements are performed ' 1l upstream and downstream of the cascade.
Mass-averaged quantities along one (two) pitch are given:
m =
ˆ s
0ρCxdy
tanα2 =
´ s0 ρCxCydy´ s
0 ρC2x dy
Yp =
´ s0 {(p01 − p02) / (p01 − p1)} ρCxdy´ s
0 ρCxdy
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
2D Blade cascade
Application to rotors and stators
Cascades are stationary ⇒ straightforward for stator blades.
For rotors, replace:α by β.~C by ~W .h0 by h0,rel .
Losses and efficiency: incompressible flow compression stage
Incompressible flow, temperature change is negligible, ρ = cte.
Upstream rotor: 1, between rotor and stator: 2, downstream stator: 3.
Actual work:∆W = h03 − h01
Minimum work required to attain same final stagnation pressure:
∆Wmin = h03ss − h01
Along p = p03, second law gives:
∆Wmin = ∆W − T∆sstage
ηtt =∆Wmin
∆W= 1−
T∆sstage
h03 − h01
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Blade loading, boundary layers and losses
Losses and efficiency: incompressible flow compression stage
Accross the rotor, h0,rel = cte:
T∆srotor =∆p0,rel
ρ=
1
2W 2
1 Yp,rotor
Accross the stator, h0 = cte:
T∆sstator =∆p0
ρ=
1
2C2
2 Yp,stator
Thus:
ηtt = 1−12
(W 2
1 Yp,rotor + C22 Yp,stator
)h03 − h01
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Blade loading, boundary layers and losses
Losses and efficiency: incompressible flow compression stage
Adverse pressure gradient: boundary layer growth(and detachment).
Wake momentum thickness θ2 correlated to diffusionon suction side.
θ2 =
ˆ s/2
−s/2
(C
C∞
)(1−
C
C∞
)dy
Diffusion factor linked to solidity:
DF =
(1−
C2
C1
)+
( |Cθ2 − Cθ1|2σC1
)
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Blade loading, boundary layers and losses
Effects of Reynolds number, Mach number and incidence
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Towards 3D: Radial equilibrium
Radial dependence & spanwise velocities
1)
U = rω ⇒ staggerangles depend on r .
2)
Bernoulli theoremaccros streamlines:
C 2θ
r=
1
ρ
∂p
∂r
For hub-to-tip ratiorh/rt . 0.8,
temporary imbalancebetween centrifugalforces and radialpressure gradients.
streamlines bend
radially until sufficient
radial transport to
recover equilibrium.
3)
Simplified radialequilibrium hypothesis:
Permanent flow;
Outside blade rows;
Cylindricalstreamtubes;
Viscous stress
neglected:
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Towards 3D: Radial equilibrium
Simplified radial equilibrium (incompressible flow)
C2θ
r=
1
ρ
dp
dr
dh = Tds −dp
ρ
1
ρ
dp0
dr=
dp
dr+ Cθ
dCθ
dr+ Cx
dCx
dr
1
ρ
dp0
dr= Cx
dCx
dr+
Cθ
r
d
dr(rCθ)
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Towards 3D: Radial equilibrium
Radial repartition of the work: vortex law
Free vortex: rCθ = cte
Constant vortex: Cθ = cte
Forced vortex: Cθ = cte · rConstant absolute angle: Cθ/Cz = cte
General vortex: Cθ = k1rn + k21r
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Secondary viscous flows
“Passage vortex”: mechanisms
Flows induced in transverse (S3) surfaces, bycreation of meridional vorticity.
Vorticity tends to conserve.
Boundary layers on hub and casing arevortical regions (vorticity ωp).
Deflection ε.
⇒ creation of a pair of passage vortices(ωs).
ωs ' 2εωp .
Other explanation based on blade-to-bladepressure gradient and streamline curvature.
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Secondary viscous flows
Other effects
Horse-Shoe vortex: induced byboundary layer impinging onleading edge.
Blade boundary layers and wakes:low momentum fluid is centrifuged(radial secondary flow).
Tip leakage vortex: induced byflow instability in the radial gapbetween blade and casing.
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Secondary viscous flows
Secondary flows, a misleading term
These mechanisms exist, butthey all non-linearly interact:
Extremely difficult to identifythem.
One dimensional Theory Design methodology for an axial-flow stage Actual 3D flows
Large scale instabilities
Stall, Stage stall and surge
Rotating stall: frequency of the order ofthe rotating frequency.Surge: system instability, slow timescales.